Universal objects in categories of reproducing kernels

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Universal objects in categories of reproducing kernelsWe continue our earlier investigation on generalized reproducing kernels, in connection with the complex geometry of $C^*$- algebra representations, by looking at them as the objects of an appropriate category. Thus the correspondence between reproducing $-*$-kernels and the associated Hilbert spaces of sections of vector bundles is made into a functor. We construct reproducing $-*$-kernels with universality properties with respect to the operation of pull-back. We show how completely positive maps can be regarded as pull-backs of universal ones linked to the tautological bundle over the Grassmann manifold of the Hilbert space $\ell^2{\mathbb N}$.